My newest Betta's fins look as though they are clumped together, with a white slimy substance attached at the ends.

Ive had other betta fish and have never experienced this problem.
It is fed bloodworms twice a day, water cleaned once a week unless its ridiculously dirty. It still has quite an appetite and is still active. Not as active as when i first got it though.

It sounds as if your fish had an infection of some sort and is shedding it's slime coat. Mine has done that a few times. Once after jumping out of her tank, and once after fighting off some mystery illness that clouded her eyes temporarily. Just keep his water warm and clean, and he'll fight off what ever it is, finish shedding the slime coat and he'll probably be fine.